This document summarizes the work of Dr. Steve Sun and his research group on computational modeling of thermo-mechanics in crystalline rock salt for nuclear waste disposal. Their work involves developing multi-scale and multi-physics models to predict the thermo-hydro-mechanical-chemical responses of salt subjected to moisture and stress. At the grain-scale, they use crystal plasticity simulations. At the aggregate scale, they model effects like grain boundary fracture. They also use computational homogenization and machine learning to develop data-driven models to predict field-scale responses while accounting for phenomena at lower scales. Their goal is to predict long-term behavior of salt-based nuclear waste repositories from first principles using a hierarchical multi-scale

1. Computational thermo-mechanics of crystalline rock
salt for nuclear waste disposal Part I: Single-crystal
simulations
1
Steve WaiChing Sun
Assistant Professor
Department of Civil Engineering and Engineering Mechanics
Columbia University

2. Acknowledgement of Current and Former Research
Group Members
2
SeonHong Na Kun Wang Nikolaos Vlassis Eric Cushman Bryant
Jinhyun Choo, PhD Yang Liu, PhD

3. Mixing granular materials
with moisture content
Motivation & Background
Artificial ground freezing
Induced Seismicity due to hydraulic
fracture, mining, CO2 storage…etc
Tire-soil interaction and off-
road mobility
Geological disposal of nuclear
waste

4. Goal: To predict THMC responses of salt
with clay and moisture additive from first
principles

5. Salt as multiphase material
Idealized intracrystalline brine inclusions and inter- crystalline pore
fluids (brine and gas) in salt (Olivella et al., 1995)

6. Governing EquationsMathematical framework: IBVP
(Represented fractured pore space)
Reconsolidated Salt as a multi-porosity materials

7. Semi-implicit FEM2 poromechanics model
Important Assumption: Effective stress principle valid for grain assemblies and REV
exists
Dry case reference : Miehe, CMAME 2003, Nitka et al, Granular Matters, 2011, Guo, Zhao,
IJNME 2014, Yang, Sun, Yuan and Fish, IJNME 2015. There are more …
Rationale: Run the least amount of RVE simulationds for each hydromechanical update.
Mixed-field
Effective stress
from unit cell
Stabilized Mixed FEM for finite strain poromechanics
model (Sun, Ostien, Salinger, IJANMG, 2013)
Strain prescribed as
BC for FEM

8. Effective Stress Principle Across Length Scale
All measureable effects of a change
of stress, such as compression,
distortion, and a change of shearing
resistance are due exclusively to
changes in effective stress.
-- Karl Terzaghi 1936
Applying this idea to get (1)
homogenized effective stress from
DEM and (2) macroscopic pore
pressure from mixed FEM.
Wang & Sun, CMAME, 2016

9. Grain-scale Modeling effort: Single-
crystal plasticity

10. Crystal Plasticity for salt

12. Elastic Predictor

13. Plastic
Corrector

15. Case 1 Case 2 Case 3

16. Comparison with experiments
0.0 2.0 4.0 6.0 8.0 10.0
0.0
5.0
10.0
15.0
20.0
25.0
Strain (%)
σ1-σ3(MPa)
Strain rate 1
25 ˚C 100 ˚C
200 ˚C 300 ˚C
400 ˚C 25 ˚C
100 ˚C 200 ˚C
300 ˚C 400 ˚C
0.0 2.0 4.0 6.0 8.0 10.0
0.0
5.0
10.0
15.0
20.0
25.0
Strain (%)
σ1-σ3(MPa)
Strain rate 2
25 ˚C 100 ˚C
200 ˚C 300 ˚C
400 ˚C 25 ˚C
100 ˚C 200 ˚C
σ1 - σ3 (w/ strain rate 1.1~1.4 x 10-7
/sec) MPa σ1 - σ3 (w/ strain rate 1.1~1.4 x 10-4
/sec) MPa

17. Aggregate-scale Modeling effort:
Boundary boundary effect

18. Salt as polycrystalline material
1. Plastic deformation, intra-granular fracture,
dislocation creeping captured by crystal plasticity.
2. Grain boundary fracture, grain boundary diffusion,
precipitation creep, recrystallization, captured by
phase field or embedded strong dis-contunities.

19. Generating traction-separation laws from
experiments and sub-scale simulations
(recursive training)
Branching strong discontinuity model
from Linder & Armero, FEAD, 2009
Details of computational
homogenization of layers see
Hirschberger, Sukumar, Steinmann,
PM, 2008
Enhanced assumed strain with branched displacement jump
Branching separation mode
Displacement
jump as input
Traction and
micro-
permeability
as output
Traction-separation law comes from sub-scale simulation

20. Computational homogenization of traction-
separation law for grain boundaries
Hill-Mandel Lemma for bulk volume
Nominal Traction
Hill-Mandel for interface
where

21. 1
2
3
4
1 2
3 4
Effective Permeability Homogenization from
deforming network models
Darcy’s Law in flow network Balance of mass for fluid network
(incompressible flow)
Network Generation Algorithm Voro++
from Chris Rycroft
Update void
topology based on
grain position
Flux from pore center I
to pore center j
Hydraulic conductivity
of edge connecting i
and j nodes

22. field-scale Modeling effort: data-driven
multiscale model through deep machine
learning

23. DRAWBACKS of hierarchical multiscale
(FEM2 or DEM-FEM) methods
1: Cost (each Gauss point requires RVE simulations)
X
2: The local DEM-FEM does not preserve length
scale. X
3. The nonlocal version requires the length scale to be
larger than the finite element (same as phase field
other nonlocal method). X

24. Proposed Solution
1: Cost (each Gauss point requires RVE simulations)
2: The local DEM-FEM does not preserve length scale.
3. The nonlocal version requires the length scale to be larger
than the finite element.
– Use offline machine learning algorithm
– Use proper regularization
– Embedded strong discontinuity or micro-polar models
to incorporate length-scale dependent information

25. Extension to multi-physics problem?
1. Topology of information becomes important.
2. The distinction among definition, governing laws and ad hoc relations is less clear.
3. How to incorporate human knowledge to speed up the training procedure?
4. How to use data-driven algorithm as a mean to carry multiscale simulations.

26. Forward Prediction and Training of Artificial Neural
Network Data-driven Law
The Biological Neuron
The artificial Neuron
 The learning rule modifies the
weights according to the input
patterns that it is presented with. In
a sense, ANNs learn by example
as do their biological counterparts.
 When the desired output are
known we have supervised
learning or learning with a teacher.
 Each neuron is a cell that uses
biochemical reactions to receive,
process and transmit information.
 When one of those neurons fire, a
positive or negative charge is
received by one of the dendrites.
 The strengths of all the received
charges are added together through
the processes of spatial and
temporal summation.
Prediction with ANN Back propagation

27. Basic Idea of data-driven modeling for solid
mechanics in a nutshell
𝝐𝒊𝒋 𝝈ij
Input layer
hidden layer(s)
Output layer
Governing equation
definition
Data-driven model
replaced material laws
1. We partition the database into training set and verification set.
2. Data of both stress and strain from experiments are used to update the
weights of nodes.
3. Error is measured by cost (objective function).
4. We find the weights of the layers such that the errors are minimized.
5. IF it works, we can use the data-driven model as a surrogate model

28. Training of Artificial Neural Network
Figure from Liu, Sun, Fish, Journal of
Applied Mechanics, 2016.
1. For classical problems, inverse problems are used to obtain optimal material
parameters.
2. For ANN data-driven model, the training set data are used to update the “weight” of
the neuron through a procedure called backward propagation.
3. The more hidden layers we put in, the more complex responses can be captured BUT
the training is harder to complete (e.g. the vanishing gradient problem).

29. Example of training sets: database for
supervised machine learning
• Can be generated from experiments and/or from microscale simulations (e.g. MD,
DEM..etc) or a combination of both but with different weight for the objective functions.

30. Incorporation of human knowledge in machine learning
Spectral ANN model
Direct ANN model
Remedy 2: Get more data with rotated frame ( cf. Lefik & Schrelfer 2003)
Common issue: ANN trained model lacks objectivity (LEFT FIGURE)
Remedy 1: we proposed – use invariants and parametrize rotations, i.e.
modify the directed graph (RIGHT RIGHT)
Example: (Lack of) Material frame indifference

31. Le Garzic et al. JSG, 2011
Field scale problem Mathematical framework: IBVP
Grain-scale DEM-network simulations
Meso-scale assumed strain poromechanics simulations
Field-scale IBVP

32. Reactivation of dual-porosity fault example
Sealing
Fault
Injection
of fluid
100 M𝑃𝑎
Drained
Drained
Drained
60 M𝑃𝑎
Drained
Idealized macroscopic dual-
porosity problem with no build-
in constitutive law
Field applications
Assumed strain embedded strong dis-
contunity problem serves as ”trainer” for
Macro-Meso ANN that generates macroscopic
dual porosity responses
DEM-network model
are serve as “trainner”
for Meso-micro ANN
that generates the
responses of joints
and micro-fracture

33. Implementation: Component-based PDE for
Dual-porosity problems (Macroscale Problem)
Wang & Sun, Proceeding of Biot’s conference, 2017
Wang & Sun, in preparation
• A sub-graph with vertices connected only by data-driven model can be identified.
• The computer model can be considered as a mathematical object called directed graph.
• Each vertex represent a physical quality
• Each edge represents a mapping or function that links the upstream and downstream
physical qualities (vertices)
Subgraph partially
or completely
replaced by ANN
models

34. Implementation: Component-based PDE with
hybrid off-line trained data-driven models
Wang & Sun, Proceeding of Biot’s conference, 2017
Wang & Sun, in preparation

35. Macroscopic Response
Macro-pore pressure Micro-pore pressure
Deviatoric strain Maximum principal stress

36. Meso-scale assumed strain response
Volumetric strain Deviatoric stress

37. Ongoing Work: Mechanics of Reconsolidated Salt
37
1. Dislocation creep vs. fluid-assisted creep
2. Grain-boundary fracture vs. intra-fracture
3. Double porosity system (grain boundary & fracture)
4. Brine inclusion
5. Pressure solution? Healing?
6. Experimental work conducted during the summer

38. Conclusion
1. We have developed a mathematical framework for ploy-
crystalline salt.
2. For each salt grain, the material response is characterized by
crystal plasticity.
3. Grain boundary is captured via assumed strain and properly
cohesive law (under development)
4. At the macroscopic level, the salt behavior is captured by
multiscale method where brine movement at different pore
systems are captured.
5. It is an ongoing work and we will incorporate the effect of
moisture, grain boundary diffusion, anisotropic fracture, other
creeping mechanism, healing …etc

39. Acknowledgements
• Army Research Office: Young Investigator Program Award
• Air Force Office of Scientific Research, Young Investigator Program Award
• National Science Foundation: EAR-1516300 and CMMI-1445033
• Sandia National Laboratories
• US Department of Energy
• Columbia University

40. THANK
YOU!
More information can be found at
http://poromehanics.weebly.com
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